The assertion (1) follows from [1]}, 2.3.7.c, since \(r=(k+1)\lambda >(k-3)\lambda \) .
The flag-transitivity of \(G\)  on \(\mathcal {D}\)  implies \(\left|G\right|=k^{2}\left|G_{x}\right|\) , \(\left|G_{x}\right|=\lambda \left( k+1\right) \left|G_{x,B}\right|\) 
and hence \(\left|G\right|<\left|G_{x}\right|^{3}\) , which
is the assertion (2).
Let \(y\)  be any point of \(\mathcal {D}\) , \(y\ne x\) , and \(B\)  be any block of \(\mathcal {D}\)  incident with \(x\) . Since \((y^{G_{x}},B^{G_{x}})\)  is a tactical
configuration by [1]}, 1.2.6, it follows that \(\left|y^{G_{x}}\right|\lambda =r\left|B\cap y^{G_{x}}\right|\) .
Hence \(\left|y^{G_{x}}\right|=(k+1)\left|B\cap y^{G_{x}}\right|\)  as \(r=(k+1)\lambda \) . This proves (3).
